Ultimate Guide to the GP Formula: Understanding its Impact and Utility
Geometric Progression (G.P.) is a fundamental concept in mathematics, particularly in the field of sequences and series. It is a type of sequence where each succeeding term is produced by multiplying each preceding term by a fixed number, known as the common ratio.
Confused! Don't worry; this article will comprehensively explain geometric progression, including its definition, formulas, and examples.
Table of Content
What is Geometric Progression?
A geometric progression (GP), also known as a geometric sequence, is a special kind of sequence where each term is obtained by multiplying the previous term by a constant number called the common ratio.
In simpler terms, imagine you have a list of numbers, and to get from one number to the next, you always multiply by the same number.
Here are some key points about geometric progressions:
- Constant ratio: The defining characteristic of a GP is the constant ratio between successive terms.
- For example, in sequences 2, 6, 18, 54, etc., each term is 3 times the one before it. So, the common ratio in this case is 3.
- Formula: The GP formula is a, ar, ar2, ar^3, ..., where:
- a is the first term.
- r is the common ratio.
- ar^n represents any term in the sequence, where n is its position (starting from 1).
- Examples:
- 1, 2, 4, 8, 16, ... (common ratio 2)
- 5, 10, 20, 40, ... (common ratio 2)
- 1/2, 1/4, 1/8, 1/16, ... (common ratio 1/2)
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Nth-Term of Geometric Progression
Let a be the first term and r be the common ratio for a geometric sequence.
Then, the second term will be: a2 = a*r
The third term will be: a3 = a*r^2
The fourth term will be: a4 = a*r^3
Similarly, the nth term will be: an = a*r^(n-1)
Therefore, the formula for the nth term of the geometric progression will be a*r^(n-1).
Examples of Geometric Progression
Example 1: Given a geometric sequence where the first term (a1) is 2, and the common ratio (r) is 3, find the first four terms of the sequence.
Answer:
First Term (a1) = 2
Second Term (a2) = 2*3^1 = 6
Third Term (a3) = 2*3^2 = 18
Fourth Term (a4) = 2*3^3 = 54
Therefore, the first four terms of the sequence are 2, 6, 18, and 54.
Example 2: Determine the fifth term of a geometric sequence where the first term is 5, and the common ratio is -2.
Answer:
As given, the first term (a1) = 5
Now, we will use the formula of the n-th term of the sequence, i.e., an = a1*r^(n-1)
=> a5 = 5*(-2)^(5-1) = 5*16 = 80
=>a5 = 80
Therefore, the 5th term of the sequence will be 80.
Example 3: In GP, a_3 = 24, and a_6 = 192. Find the first term (a1) and common ratio (r).
Answer:
Here, we have a3 = 24 and a6 = 192, and we know the formula of the nth term of GP, i.e., an = a1*r^(n-1)
=> a3 = a1*r^2 = 24 and a6 = a1*r^5 = 192
=> a6/a3 = (a1*r^5)/(a1*r^2) = 192/24
=> r^3 = 8
=> r = 2
Since we have found the value of r, so by substituting the value of r in the formula of a3 or a6, we will get the first term.
As 24 = a3 = a1*(2)^2
=> 4*a1 = 24
=> a1 = 6
therefore, from above, we get:
first term (a1) = 6 and
common ratio (r) = 2.
Properties of Geometric Progression
Basic Properties:
- Constant Ratio: Each term equals the previous term multiplied by a fixed non-zero number called the common ratio (r).
- Product of equidistant terms: In a finite GP, the product of terms equidistant from the beginning and end is the same.
- For example, in sequences 2, 6, 18, and 54, the product of 2 and 54 (first and last term) equals the product of 6 and 18 (second and third term).
- Zero terms: A GP cannot have any zero terms, as multiplying by zero would break the constant ratio rule.
Transformations:
- Multiplication/Division by constant: Multiplying or dividing each term by a non-zero constant results in a new GP with the same common ratio.
- Reciprocals: The reciprocals of all terms in a GP also form a GP with the reciprocal of the original common ratio.
- Raising to a power: If each term is raised to the same non-zero power, the resulting sequence is also a GP, with the common ratio raised to the same power.
Relationships between terms:
- Three-term test: Three non-zero terms a, b, and c are in GP if and only if b² = ac.
- nth term formula: The value of any term (n) in a GP can be calculated using the formula an = a * r^(n-1), where a is the first term and r is the common ratio.
Other Properties:
- Convergence/Divergence: A GP with |r| < 1 converges to 0 as n approaches infinity.
- A GP with |r| > 1 diverges (grows or shrinks without bound).
- A GP with r = 1 remains constant.
- The sum of infinite GP: The sum of an infinite GP with |r| < 1 is given by S = a / (1-r), where a is the first term and r is the common ratio.
Types of Geometric Progression
GP (or Geometric Progression) are broadly classified into 2 different categories:
Finite Geometric Progression
This type of GP has a fixed and specific number of terms. Common examples include:
- 2, 6, 18, 54,.... (common ratio = 3)
- 1/2, 1/4, 1/8, 1/16, .... (common ratio = 1/2)
The sum of Fininte GP = a(1-r^n) / 1-r, for r not equal to 1.
Infinite Geometric Progression
This type of GP has an unlimited number of terms, i.e., counting forever. Examples include
- 1, 2, 4, 8, 16, .....
- 1 - 1/3 + 1/9 - 1/27 + ...
Infinite GP has different behavior regarding the convergence and divergence depending on the value of the common ratio:
- Convergent Infinite GP: Infinite GP is convergent if the absolute value of the common ratio is less than 1, i.e., |r|< 1. In this case, the sum of the infinite gp is calculated with the formula: S = a/1-r
- Divergent Infinite GP: Infinite GP is divergent if the absolute value of the common ratio is greater than or equal to 1.
Conclusion
Geometric progression is a fundamental mathematical concept characterized by a sequence of numbers. Each term is obtained by multiplying the preceding term by a fixed, non-zero number known as the common ratio. It has real-world applications, such as calculating population growth and interest. Understanding its definition, formulas, and properties enables effective analysis and problem-solving.
This article provides a comprehensive overview, including its definition, key formulas, properties, and examples, equipping readers with a solid understanding.
In the following articles, we will explore how to find the sum of GP (i.e., the sum of the GP formula) and the difference between the AP and GP formula.
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FAQs on GP Formula
What is a geometric progression and how is it different from an arithmetic progression?
A geometric progression (GP) is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio.
For example, in the sequence 2, 4, 8, 16, ..., each term is obtained by multiplying the previous term by 2, making the common ratio 2.
The difference between a geometric progression and an arithmetic progression (AP) lies in how the sequences are generated:
- In an Arithmetic Progression, each term is obtained by adding a constant value, known as the common difference, to the previous term. For instance, in the sequence 2, 5, 8, 11, ..., each term is obtained by adding 3 to the previous term.
- In a Geometric Progression, each term is obtained by multiplying the previous term by a constant value, known as the common ratio, as explained above.
How do you find the nth term of a geometric progression?
- The GP formula is a, ar, ar^2, ar^3, ..., where:
- a is the first term.
- r is the common ratio.
- ar^n represents any term in the sequence, where n is its position (starting from 1).
What are some real-world applications of geometric progressions?
Real-world applications of geometric progressions include:
- Compound Interest: The amount of money grows by a fixed percentage each period (year, month, etc.), which a geometric sequence can model.
- Population Growth: Populations of organisms can grow exponentially if each individual produces more than one offspring, which a geometric sequence can represent.
- Physics and Engineering: Certain phenomena, like the decay of radioactive substances or the charging and discharging processes of capacitors in electronics, follow patterns described by geometric progressions.
- Computer Science: In algorithms, especially those involving divide and conquer strategies, geometric sequences can often model the complexity or the number of operations.
Vikram has a Postgraduate degree in Applied Mathematics, with a keen interest in Data Science and Machine Learning. He has experience of 2+ years in content creation in Mathematics, Statistics, Data Science, and Mac... Read Full Bio
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